91Ó°ÊÓ

Myers-Briggs: Actors Isabel Myers was a pioneer in the study of personality types. The following information is taken from A Guide to the Development and Use of the Myers-Briggs Type Indicator by Myers and McCaulley (Consulting Psychologists Press). In a random sample of 62 professional actors, it was found that 39 were extroverts. (a) Let \(p\) represent the proportion of all actors who are extroverts. Find a point estimate for \(p\). (b) Find a \(95 \%\) confidence interval for \(p .\) Give a brief interpretation of the meaning of the confidence interval you have found. (c) Check Requirements Do you think the conditions \(n p>5\) and \(n q>5\) are satisfied in this problem? Explain why this would be an important consideration.

Step by step solution

01

Given Information and Definitions

We have a random sample of 62 actors, with 39 identified as extroverts. Here, \( n = 62 \) (total sample size) and \( x = 39 \) (number of extroverts). \( p \) is defined as the proportion of all actors who are extroverts.

02

Calculate the Point Estimate for p

The point estimate for \( p \), the sample proportion, is given by the formula:\[p = \frac{x}{n} = \frac{39}{62}.\]Calculating this gives:\[p = 0.629 \, \text{(rounded to three decimal places)}.\]

03

Confidence Interval Formula for Proportion

The formula for a \(95\%\) confidence interval for a proportion is:\[p \pm z \sqrt{\frac{p(1-p)}{n}},\]where \( z \) is the z-score corresponding to the desired confidence level. For \(95\%\), \( z \approx 1.96 \).

04

Calculate the Confidence Interval

First, compute the standard error:\[SE = \sqrt{\frac{0.629 (1-0.629)}{62}} = 0.0622 \, \text{(rounded to four decimal places)}.\]Now, calculate the margin of error:\[ME = 1.96 \times 0.0622 = 0.1219 \, \text{(rounded to four decimal places)}.\]The confidence interval is:\[0.629 \pm 0.1219 = (0.5071, 0.7509).\]

05

Interpretation of the Confidence Interval

The 95% confidence interval \((0.5071, 0.7509)\) suggests that we can be 95% confident that the true proportion of all actors who are extroverts is between \(50.71\%\) and \(75.09\%\).

06

Check Requirements np>5 and nq>5

First, compute \(np\) and \(nq\):\[np = 62 \times 0.629 = 39.0, \ nq = 62 \times (1-0.629) = 23.0.\]Both \( np = 39.0 \) and \( nq = 23.0 \) are greater than 5, therefore, the conditions are satisfied. This ensures that the sample size is sufficient for the confidence interval to be valid.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval

A confidence interval provides a range of values that is likely to contain the true population parameter. In our exercise, it helps us estimate the proportion of actors who are extroverts. It's calculated using a known confidence level, such as 95%. This means we are 95% confident that our interval truly contains the population parameter.
For the calculation, we use the formula:

• Point Estimate ± Margin of Error
• The Margin of Error is calculated using the standard error and a z-score based on our confidence level

The interpretation of a 95% confidence interval is that if we were to take 100 different samples and compute this interval each time, 95 of the intervals would contain the true proportion of extroverted actors.

Point Estimate

A point estimate is a single value that serves as an estimate of a population parameter. In this case, it's used to estimate the proportion of extroverted actors in the entire population.
The point estimate is calculated using the sample data with the formula:

• Sample Proportion (\(p\)) = \(\frac{x}{n}\), where \(x\) is the number of successes, and \(n\) is the sample size.

For our exercise, 39 out of a sample of 62 actors were extroverts, leading to a point estimate of 0.629. This is our best guess of the true proportion based on the sample data.

Sample Proportion

The sample proportion is the ratio of the number of successes in a sample to the total sample size. In our example, it's the proportion of extroverts in the group of actors surveyed.
It is computed using the formula:

• \(p = \frac{x}{n}\)

The sample proportion helps give an idea of what the true proportion might be if we could survey the entire population of actors. For our problem, the sample proportion is 0.629, meaning that about 62.9% of the actors in our sample are extroverted, serving as a stepping stone for further statistical analysis like forming a confidence interval.

Conditions for Validity

Before trusting the results of our confidence interval, it's crucial to check if certain conditions for validity are met. These conditions ensure that the statistical methods used to create the confidence interval are appropriate.
The key requirements are:

• \(np > 5\)
• \(nq > 5\), where \(q = 1 - p\)

These inequalities must be satisfied to assume a normal distribution, which is necessary for accurate confidence interval estimation. In the exercise, both conditions are satisfied because 39 (\(np\)) and 23 (\(nq\)) are both greater than 5, confirming that our sampling method is sound.